Nonconvex Mathematical Programming Research Articles

A characterization of stationary Nash equilibria of constrained stochastic games with independent state processes

We characterize the stationary Nash equilibria of a N-player general sum constrained stochastic game with independent state processes studied in Altman et al. (2008), via the global minimizers of a certain non-convex mathematical program. We identify a subclass of these N-player general sum games called Additive Cost—Decoupled Constraints Cost games for which the above characterization is via linear programs. The linear programs for two player zero sum games as given in Altman et al. (2005) can be obtained as a special case of our mathematical program.

The [formula omitted] exact [formula omitted]-penalty function method and [formula omitted]-invex mathematical programming problems

In this paper, we consider G -invex mathematical programming problems and show the efficiency of G -invexity notion in proving optimality results for such nonconvex optimization problems. Further, we introduce a new exact penalty function method, called the l 1 exact G -penalty function method and use it to solve nonconvex mathematical programming problems with G -invex functions. In this method, the so-called exact G -penalized optimization problem associated with the original optimization problem is constructed. The equivalence between the sets of optimal solutions of the original mathematical programming problem and of its associated G -penalized optimization problem is established under suitable G -invexity assumptions. Also lower bounds on the penalty parameter are given, for which above this result is true. It turns out that, for some nonconvex optimization problems, it is not possible to prove the same result for the classical l 1 penalty function method under invexity assumption.

A new exact exponential penalty function method and nonconvex mathematical programming

A new exact penalty function method, called the l 1 exact exponential penalty function method, is introduced. In this approach, the so-called the exponential penalized optimization problem with the l 1 exact exponential penalty function is associated with the original optimization problem with both inequality and equality constraints. The l 1 exact exponential penalty function method is used to solve nonconvex mathematical programming problems with r-invex functions (with respect to the same function η). The equivalence between sets of optimal solutions of the original mathematical programming problem and of its associated exponential penalized optimization problem is established under suitable r-invexity assumption. Also lower bounds on the penalty parameter are given, for which above these values, this result is true.

THE l1 PENALTY FUNCTION METHOD FOR NONCONVEX DIFFERENTIABLE OPTIMIZATION PROBLEMS WITH INEQUALITY CONSTRAINTS

In this paper, some new results on the l1 exact penalty function method are presented. A simple optimality characterization is given for the nonconvex differentiable optimization problems with inequality constraints via the l1 exact penalty function method. The equivalence between sets of optimal solutions in the original mathematical programming problem and its associated exact penalized optimization problem is established under suitable r-invexity assumption. The penalty parameter is given, above which this equivalence holds. Furthermore, the equivalence between a saddle point in the considered nonconvex mathematical programming problem with inequality constraints and a minimizer in its penalized optimization problem with the l1 exact penalty function is also established.

On computational search for optimistic solutions in bilevel problems

The linear-linear and quadratic-linear bilevel programming problems are considered. Their optimistic statement is reduced to a nonconvex mathematical programming problem with the bilinear structure. Approximate algorithms of local and global search in the obtained problems are proposed. The results of computational solving randomly generated test problems are given and analyzed.

Local search in a quadratic-linear bilevel programming problem

A quadratic-linear bilevel programming problem is considered. Its optimistic statement is reduced to a nonconvex mathematical programming problem with a quadratic-bilinear structure. An approximate algorithm of a local search in the problem obtained is proposed, proved, and tested.

Delay QoS and MAC Aware Energy-Efficient Data-Aggregation Routing in Wireless Sensor Networks

By eliminating redundant data flows, data aggregation capabilities in wireless sensor networks could transmit less data to reduce the total energy consumption. However, additional data collisions incur extra data retransmissions. These data retransmissions not only increase the system energy consumption, but also increase link transmission delays. The decision of when and where to aggregate data depends on the trade-off between data aggregation and data retransmission. The challenges of this problem need to address the routing (layer 3) and the MAC layer retransmissions (layer 2) at the same time to identify energy-efficient data-aggregation routing assignments, and in the meantime to meet the delay QoS. In this paper, for the first time, we study this cross-layer design problem by using optimization-based heuristics. We first model this problem as a non-convex mathematical programming problem where the objective is to minimize the total energy consumption subject to the data aggregation tree and the delay QoS constraints. The objective function includes the energy in the transmission mode (data transmissions and data retransmissions) and the energy in the idle mode (to wait for data from downstream nodes in the data aggregation tree). The proposed solution approach is based on Lagrangean relaxation in conjunction with a number of optimization-based heuristics. From the computational experiments, it is shown that the proposed algorithm outperforms existing heuristics that do not take MAC layer retransmissions and the energy consumption in the idle mode into account.

Penalty function methods and a duality gap for invex optimization problems

In this paper, the penalty function method is used to study duality in nonconvex mathematical programming problems. In particular, we prove the zero duality gap between optimization problems involving invex functions with respect to the same function η and their Lagrangian dual problems. The results proved in the paper are illustrated by suitable examples of nonconvex optimization problems.

Generalization of the newton method for one class of nonconvex mathematical programming problems

This paper continues the study commenced in [1] and [2], where we consider iterative procedures, finding stationary points of smooth functions on a class of nonconvex sets. We generalize the Newton method applied for the solution of convex programming problems for the case, when constraints are represented as a set-theoretical difference of a convex set and a union of several convex sets. We formulate and prove a proposition on the convergence of the algorithm.

Generalization of the newton method for one class of nonconvex mathematical programming problems

Generalization of the newton method for one class of nonconvex mathematical programming problems

A nonlinear nonconvex minimum total heat transfer area formulation for ocean thermal energy conversion (OTEC) systems

In this paper a technical analysis of an ocean thermal energy conversion (OTEC) system is performed. Specifically, we present a general mathematical framework for the synthesis of OTEC power generating systems. The overall synthesis task is to minimize heat exchange area requirements, while generating some fraction of the maximum net power recoverable from hot and cold ocean water. The resulting problem formulation yields a nonlinear, nonconvex mathematical program; however, we show that globally optimal solutions for this program are easily obtained explicitly through a direct optimization approach with minimal computational effort over a wide range of thermodynamic conditions. The proposed analysis is demonstrated on a case study involving the generation of hydrogen by an OTEC system with a pure ammonia working fluid.

Partially B-Regular Optimization and Equilibrium Problems

This paper introduces a concept termed partial B-regularity for a feasible solution to a bivariate constraint system and shows that this condition leads to the equivalence between the B-stationarity of a pair of lifted and unlifted programs. In particular, for an optimization problem with a univariate pseudoconvex objective function constrained by such a nonconvex bivariate system, partial B-regularity provides a sufficient condition for a B-stationary point to be globally optimal. Applications of partial B-regularity to several classes of optimization and equilibrium problems are presented; these include a lexicographic optimization problem, a nonconvex mathematical program with equilibrium constraints (MPEC) that arises from a convex implicit value-function optimization problem, and a Nash equilibrium program with equilibrium constraints.

Interactive balance space approach for solving multi-level multi-objective programming problems

This paper studies a multi-level multi-objective decision-making (ML-MODM) problems with linear or non-linear constraints. The objective functions at each level are non-linear functions, which are to be maximized or minimized. This paper presents a three-level multi-objective decision-making (TL-MODM) model and an interactive algorithm for solving such a model. The algorithm simplifies three-level multi-objective decision-making problems by transforming them into separate multi-objective decision making problems at each level, thereby avoiding the difficulty associated with non-convex mathematical programming. Our algorithm is an extension of the work of Shi and Xia [X. Shi, H. Xia, Interactive bi-level multi-objective decision making, Journal of the Operational Research Society 48 (1997) 943–949], which dealt with interactive bi-level multi-objective decision-making problems, with some modifications in assigning satisfactoriness to each objective function in all the levels of the TL-MODM problem. Also, we solve each separate multi-objective decision making problem of the TL-MODM problem by the balance space approach. A new formula is introduced to interconnect the satisfactoriness and the proportions of deviation needed to reflect the relative importance of each objective function. Thus, we have the proportions of deviation including satisfactoriness. In addition, we present new definitions for the satisfactoriness and the preferred solution in view of singular-level multi-objective decision making problems that corresponds to the η-optimal solution of the balance space approach. Also, new definitions for the feasible solution and the preferred solution ( η-optimal point) of the TL-MODM problem are presented. An illustrative numerical example is given to demonstrate the algorithm.

Kuhn–Tucker sufficiency for global minimum of multi-extremal mathematical programming problems

The Kuhn–Tucker Sufficiency Theorem states that a feasible point that satisfies the Kuhn–Tucker conditions is a global minimizer for a convex programming problem for which a local minimizer is global. In this paper, we present new Kuhn–Tucker sufficiency conditions for possibly multi-extremal nonconvex mathematical programming problems which may have many local minimizers that are not global. We derive the sufficiency conditions by first constructing weighted sum of square underestimators of the objective function and then by characterizing the global optimality of the underestimators. As a consequence, we derive easily verifiable Kuhn–Tucker sufficient conditions for general quadratic programming problems with equality and inequality constraints. Numerical examples are given to illustrate the significance of our criteria for multi-extremal problems.

A practical approach to approximate bilinear functions in mathematical programming problems by using Schur's decomposition and SOS type 2 variables

This paper provides a new methodology to solve bilinear, non-convex mathematical programming problems by a suitable transformation of variables. Schur's decomposition and special ordered sets (SOS) type 2 constraints are used resulting in a mixed integer linear or quadratic program in the two applications shown. While Beale, Tomlin and others developed the use of SOS type 2 variables to handle non-convexities, our approach is novel in two aspects. First, the use of Schur's decomposition as an integral part of the approximation step is new and leads to a numerically viable method to separate the variables. Second, the combination of our approach for handling bilinear side constraints in a complementarity or equilibrium problem setting is also new and opens the way to many interesting and realistic modifications to such models. We contrast our approach with other methods for solving bilinear problems also known as indefinite quadratic programs. From a practical point of view our methodology is helpful since no specialized procedures need to be created so that existing solvers can be used. The approach is illustrated with two engineering examples and the mathematical analysis appears in the Appendices.

A New Mathematical-Programming Framework for Facility-Layout Design

We present a new framework for efficiently finding competitive solutions for the facility-layout problem. This framework is based on the combination of two new mathematical-programming models. The first model is a relaxation of the layout problem and is intended to find good starting points for the iterative algorithm used to solve the second model. The second model is an exact formulation of the facility-layout problem as a nonconvex mathematical program with equilibrium constraints (MPEC). Aspect ratio constraints, which are frequently used in facility-layout methods to restrict the occurrence of overly long and narrow departments in the computed layouts, are easily incorporated into this new framework. Finally, we present computational results showing that the complete framework can be solved efficiently using widely available optimization software, and the resulting layouts improve on those obtained using previous approaches in the literature. Moreover, the framework can be used to find different competitive layouts with relatively little computational effort, which is advantageous for a user who wishes to consider several competitive layouts rather than simply using a mathematically optimal layout.

A goal-seeking approach to capital budgeting

This paper articulates how a goal-seeking model addresses a variation of the capital-budgeting problem. Focused toward technology modernization in the public sector, this multi-criteria optimization model explicitly considers the diverse functions of the organization. In particular, the synergism amongst the functions is modeled as a multiplicative value function. The model is solved by the “constraint reduced feasible-region method”, resulting in a non-convex mathematical program that produces numerical intricacies. Linearization of the criterion (objective) functions reduces such intricacy. An Army-modernization acquisition-study was used to illustrate the proposed model, showing that its non-inferior solutions are remarkably stable. Comparison is also made with other approaches, typically formulated as goal-setting programs. The model highlights how technology acquisitions are affected as the priorities of each organizational function changes.

Computation of Bounds on Population Parameters When the Data Are Incomplete

This paper continues our research on the identification and estimation of statistical func- tionals when the sampling process produces incomplete data due to missing observations or interval measurement of variables. Incomplete data usually cause population parameters of interest in appli- cations to be unidentified except under untestable and often controversial assumptions. However, it is often possible to identify sharp bounds on these parameters. The bounds are functionals of the population distribution of the available data and do not rely on untestable assumptions about the process through which data become incomplete. They contain all logically possible values of the population parameters. Moreover, every parameter value within the bounds is consistent with some model of the process that generates incomplete data. The bounds can be estimated consistently by replacing the population distribution of the data with the empirical distribution in the functionals that give the bounds. In practice, this is straightforward in some circumstances but computationally bur- densome in others; in general, the bounds are the solutions to non-convex mathematical programming problems that can be difficult to solve. Horowitz and Manski (Censoring of Outcomes and Regressors Due to Survey Nonresponse: Identification and Estimation Using Weights and Imputations, Journal of Econometrics 84 (1998), pp. 37-58; Nonparametric Analysis of Randomized Experiments with Missing Covariate and Outcome Data, Journal of the American Statistical Association 95 (2000), pp. 77-84) studied nonparametric mean regression with missing data. In this paper, we first describe the general problem. We then present new findings on the computation of bounds on best linear predictors under square loss. We describe a genetic algorithm to compute sharp bounds and a min- imax approach yielding simple but non-sharp outer bounds. We use actual data to demonstrate the computations.

Stability and Duality of Nonconvex Problems via Augmented Lagrangian

A nonconvex mathematical programming problem on a topological linear space in the presence of an inequality constraint is considered. An augmented Lagrangian function is constructed by using supporting cones to the epigraph of a usual perturbation function. Extremal values of prime and dual problems are shown to be equivalent under the stability conditions derived.

Convexity of Nonlinear Image of a Small Ball with Applications to Optimization

Let f : X → Y be a nonlinear differentiable map, X, Y are Hilbert spaces, B(a, r) is a ball in X with a center a and radius r. Suppose f � (x) is Lipschitz in B(a, r) with Lipschitz constant L and f � (a) is a surjection: f � (a)X = Y ; this implies the existence of ν> 0s uch that ||f � (a) ∗ y|| ν||y||, ∀y ∈ Y. Then, if e< min{r, ν/(2L)} ,t he imageF = f( B(a, e))of the ball B(a, e) is convex. This result has numerous applications in optimization and control. First, duality theory holds for nonconvex mathematical programming problems with extra constraint ||x − a|| e. Special effective algorithms for such optimization problems can be constructed as well. Second, the reachability set for 'small power control' is convex. This leads to various results in optimal control.